Optimal. Leaf size=17 \[ -\frac {1}{7 c^3 e (d+e x)^7} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{7 c^3 e (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)^8} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^8} \, dx}{c^3}\\ &=-\frac {1}{7 c^3 e (d+e x)^7}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{7 c^3 e (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 16, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{7 c^{3} e \left (e x +d \right )^{7}}\) | \(16\) |
norman | \(-\frac {1}{7 c^{3} e \left (e x +d \right )^{7}}\) | \(16\) |
gosper | \(-\frac {1}{7 \left (e x +d \right ) e \,c^{3} \left (e^{2} x^{2}+2 d x e +d^{2}\right )^{3}}\) | \(34\) |
risch | \(-\frac {1}{7 e \left (e x +d \right )^{3} c^{3} \left (e^{2} x^{2}+2 d x e +d^{2}\right )^{2}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (15) = 30\).
time = 0.27, size = 97, normalized size = 5.71 \begin {gather*} -\frac {1}{7 \, {\left (c^{3} x^{7} e^{8} + 7 \, c^{3} d x^{6} e^{7} + 21 \, c^{3} d^{2} x^{5} e^{6} + 35 \, c^{3} d^{3} x^{4} e^{5} + 35 \, c^{3} d^{4} x^{3} e^{4} + 21 \, c^{3} d^{5} x^{2} e^{3} + 7 \, c^{3} d^{6} x e^{2} + c^{3} d^{7} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (15) = 30\).
time = 3.54, size = 97, normalized size = 5.71 \begin {gather*} -\frac {1}{7 \, {\left (c^{3} x^{7} e^{8} + 7 \, c^{3} d x^{6} e^{7} + 21 \, c^{3} d^{2} x^{5} e^{6} + 35 \, c^{3} d^{3} x^{4} e^{5} + 35 \, c^{3} d^{4} x^{3} e^{4} + 21 \, c^{3} d^{5} x^{2} e^{3} + 7 \, c^{3} d^{6} x e^{2} + c^{3} d^{7} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (15) = 30\).
time = 0.24, size = 112, normalized size = 6.59 \begin {gather*} - \frac {1}{7 c^{3} d^{7} e + 49 c^{3} d^{6} e^{2} x + 147 c^{3} d^{5} e^{3} x^{2} + 245 c^{3} d^{4} e^{4} x^{3} + 245 c^{3} d^{3} e^{5} x^{4} + 147 c^{3} d^{2} e^{6} x^{5} + 49 c^{3} d e^{7} x^{6} + 7 c^{3} e^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 15, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-1\right )}}{7 \, {\left (x e + d\right )}^{7} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 105, normalized size = 6.18 \begin {gather*} -\frac {1}{7\,c^3\,d^7\,e+49\,c^3\,d^6\,e^2\,x+147\,c^3\,d^5\,e^3\,x^2+245\,c^3\,d^4\,e^4\,x^3+245\,c^3\,d^3\,e^5\,x^4+147\,c^3\,d^2\,e^6\,x^5+49\,c^3\,d\,e^7\,x^6+7\,c^3\,e^8\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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